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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemeiota | Structured version Visualization version GIF version |
Description: A translation is uniquely determined by one of its values. (Contributed by NM, 18-Apr-2013.) |
Ref | Expression |
---|---|
cdlemg1c.l | ⊢ ≤ = (le‘𝐾) |
cdlemg1c.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg1c.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg1c.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemeiota | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2822 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → (𝐹‘𝑃) = (𝐹‘𝑃)) | |
2 | simp3 1134 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝑇) | |
3 | cdlemg1c.l | . . . . . . 7 ⊢ ≤ = (le‘𝐾) | |
4 | cdlemg1c.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdlemg1c.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdlemg1c.t | . . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | 3, 4, 5, 6 | ltrnel 37274 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
8 | 7 | 3com23 1122 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
9 | 3, 4, 5, 6 | cdleme 37695 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) |
10 | 8, 9 | syld3an3 1405 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) |
11 | fveq1 6668 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑃) = (𝐹‘𝑃)) | |
12 | 11 | eqeq1d 2823 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑃) = (𝐹‘𝑃) ↔ (𝐹‘𝑃) = (𝐹‘𝑃))) |
13 | 12 | riota2 7138 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) → ((𝐹‘𝑃) = (𝐹‘𝑃) ↔ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) = 𝐹)) |
14 | 2, 10, 13 | syl2anc 586 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) = (𝐹‘𝑃) ↔ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) = 𝐹)) |
15 | 1, 14 | mpbid 234 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) = 𝐹) |
16 | 15 | eqcomd 2827 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∃!wreu 3140 class class class wbr 5065 ‘cfv 6354 ℩crio 7112 lecple 16571 Atomscatm 36398 HLchlt 36485 LHypclh 37119 LTrncltrn 37236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-riotaBAD 36088 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-undef 7938 df-map 8407 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-llines 36633 df-lplanes 36634 df-lvols 36635 df-lines 36636 df-psubsp 36638 df-pmap 36639 df-padd 36931 df-lhyp 37123 df-laut 37124 df-ldil 37239 df-ltrn 37240 df-trl 37294 |
This theorem is referenced by: cdlemg1cN 37722 cdlemg1cex 37723 cdlemm10N 38253 |
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